where \(\alpha _{i}\)  and \(\beta _{i}\)  are constants, and \(\alpha _{i}=\beta _{i}=0\)  corresponds to the Einstein-Hilbert action. The special case \(\alpha _{2}\ne 0, \alpha _{i} =0(i\ne 2)\) , and \(\beta _{i} = 0\)  is \(R^{2}\)  gravity and known as the Starobinsky model [1]}.
In \(f(R)\)  gravity, the interaction of spacetime and matter is different from GR.
In GR, gravity appears as the curvature of spacetime, where the source of this curvature is all forms of matter. In the absence of any mass or energy, spacetime can become completely flat.
What \(f(R)\)  gravity does is allow spacetime to act as a source of its curvature so that there can still be some curvature even if spacetime is empty of matter. Therefore, as the universe expands and the matter density decreases, some curvature may remain that is capable of driving cosmic acceleration.
